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London Mathematical Society Lecture Note Series: 380

Real and Complex

Singularities

Edited by

M. MANOEL

Universidade de Sao Paulo, Brazil

M. C. ROMERO FUSTER

Universitat de Valencia, Spain

C. T. C. WALL

University of Liverpool, UK

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no reproduction of any part may take place without the written

permission of Cambridge University Press.

First published 2010

Printed in the United Kingdom at the University Press, Cambridge

A catalogue record for this publication is available from the British Library

Library of Congress Cataloging in Publication Data

International Workshop on Real and Complex Singularities (10th : 2008 : Sao Carlos,

Sao Paulo, Brazil)

Real and complex singularities / edited by M. Manoel, M. C. Romero Fuster, C. T. C Wall.

p. cm. – (London Mathematical Society lecture note series ; 380)

Includes bibliographical references.

ISBN 978-0-521-16969-1 (pbk.)

1. Singularities (Mathematics) – Congresses. I. Manoel, M., 1966– II. Romero Fuster,

M. C. III. Wall, C. T. C. (Charles Terence Clegg) IV. Title. V. Series.

QA614.58.I527 2008

516.3′5 – dc22 2010030522

ISBN 978-0-521-16969-1 Paperback

Cambridge University Press has no responsibility for the persistence or

accuracy of URLs for external or third-party internet websites referred to in

this publication, and does not guarantee that any content on such websites is,

or will remain, accurate or appropriate.

This volume is dedicated to

Maria Aparecida Ruas

and

Terence Gaffney

on their 60th birthdays

Maria Aparecida Ruas

Terence Gaffney

Contents

Preface page xiii

1 On a conjecture by A. Durfee E. Artal Bartolo,

J. Carmona Ruber & A. Melle-Hernandez 1

1 Introduction 12 Invariants of singularities 23 Superisolated surface singularities 54 Zariski pairs 11

References 15

2 On normal embedding of complex algebraic surfaces

L. Birbrair, A. Fernandes & W. D. Neumann 17

1 Introduction 172 Proof 18

References 21

3 Local Euler obstruction, old and new, II Jean-Paul

Brasselet & Nivaldo G. Grulha Jr. 23

1 Introduction 232 Definition of the Euler obstruction 263 Local Euler obstruction for a frame 294 Euler obstruction and hyperplane sections 305 The local Euler obstruction of a function 326 The Euler obstruction and the Euler defect 337 The Euler defect at general points 348 The Euler obstruction via Morse theory 349 Generalizations of Milnor number 3510 Bruce and Roberts’ Milnor Number 3611 Local Euler obstruction of an analytic map 38

vii

viii Contents

12 Local Euler obstruction of 1-forms 3913 Local Chern obstruction of collections of 1-forms and

special points 41References 43

4 Branching of periodic orbits in reversible Hamiltonian

systems C. A. Buzzi, L. A. Roberto & M. A. Teixeira 46

1 Introduction 462 Preliminaries 493 Belitskii normal form 554 Liapunov–Schmidt reduction 565 Birkhoff normal form 596 Two degrees of freedom 607 Three degrees of freedom 658 Examples 68

References 69

5 Topological invariance of the index of a binary differential

equation L. S. Challapa 71

1 Introduction 712 Index of a binary differential equation 723 Topological invariance of I (E, 0) 754 Examples 79

References 80

6 About the existence of Milnor fibrations

J. L. Cisneros-Molina & R. N. Araujo Dos Santos 82

1 Introduction 822 Milnor fibrations for complex maps 863 Milnor fibrations for real maps 91

References 101

7 Counting hypersurfaces invariant by one-dimensional

complex foliations Maurıcio Correa Jr. & Marcio

G. Soares 104

1 Introduction 1042 Statement of results 1063 Examples 1094 Proofs 110

References 113

8 A note on topological contact equivalence J. C. F. Costa 114

1 Introduction 1142 Topological contact equivalence 115

Contents ix

3 Properties and invariants 115References 123

9 Bi-Lipschitz equivalence, integral closure

and invariants Terence Gaffney 125

1 Introduction 1252 The Lipschitz saturation of an ideal or module 1263 Bi-Lipschitz equisingularity of families of curves 132

References 136

10 Solutions to PDEs and stratification conditions

Terence Gaffney 138

1 Introduction 1382 The method of characteristics and W stratifications 138

References 145

11 Real integral closure and Milnor fibrations

Terence T. Gaffney & Raimundo Araujo Dos Santos 146

1 Introduction 1462 Notations and setup 1493 Some results 150

References 156

12 Surfaces around closed principal curvature lines,

an inverse problem R. Garcia, L. F. Mello &

J. Sotomayor 158

1 Introduction 1582 Preliminary results 1603 Hyperbolic principal cycles 1624 Concluding remarks 165

References 166

13 Euler characteristics and atypical values

Helmut A. Hamm 167

1 Cohomological results 1682 Absence of vanishing cycles 1743 Topological triviality 178

References 183

14 Answer to a question of Zariski A. Hefez &

M. E. Hernandes 185

1 Introduction 1852 Classification of the equisingularity class 〈7, 8〉 1853 The question of Zariski 186

References 189

x Contents

15 Projections of timelike surfaces in the de Sitter

space Shyuichi Izumiya & Farid Tari 190

1 Introduction 1902 Preliminaries 1913 The family of projections along geodesics in Sn

1 1954 Projections of timelike surfaces in S3

1 1985 Duality 2056 Appendix 208

References 209

16 Spacelike submanifolds of codimension at most two in de

Sitter space M. Kasedou 211

1 Introduction 2112 Lightcone Gauss images of spacelike hypersurfaces 2123 Lightcone height functions 2154 Generic properties and de Sitter hyperhorospheres 2175 Spacelike surfaces in de Sitter 3-space 2206 Spacelike submanifolds of codimension 2 2217 Lightlike hypersurfaces and contact with lightcones 2238 Lightcone Gauss maps and lightcone height functions 2249 Classification in de Sitter 4-space 226

References 227

17 The geometry of Hopf and saddle-node bifurcations for

waves of Hodgkin-Huxley type Isabel S. Labouriau &

Paulo R. F. Pinto 229

1 Introduction – equations of Hodgkin-Huxley type 2302 Linearisation 2333 The exchange of stability set 2354 The exchange of stability set in (HHW) 2385 Example: single channel Hodgkin-Huxley equations 241

References 244

18 Global classifications and graphs J. Martınez-Alfaro,

C. Mendes de Jesus & M. C. Romero-Fuster 246

1 Introduction 2462 Surfaces and graphs 2473 3-Manifolds and graphs 262

References 264

19 Real analytic Milnor fibrations and a strong Łojasiewicz

inequality David B. Massey 268

1 Introduction 2682 Singular values of matrices 271

Contents xi

3 Ł-maps and Ł-weights 2744 Milnor’s conditions (a) and (b) 2825 The main theorem 2866 Comments and questions 290

References 291

20 An estimate of the degree of L-determinacy by the degree

of A-determinacy for curve germs T. Nishimura 293

1 Introduction 2932 Proof of Theorem 1.1 294

References 297

21 Regularity of the transverse intersection of two regular

stratifications Patrice Orro & David Trotman 298

1 Introduction 2982 Regular stratifications 299

References 303

22 Pairs of foliations on surfaces Farid Tari 305

1 Introduction 3052 Examples from differential geometry 3073 Classification 3114 Invariants 3255 Bifurcations 3306 Appendix 331

References 334

23 Bi-Lipschitz equisingularity David Trotman 338

1 An oft-heard slogan 3382 Existence of Whitney and Verdier stratifications 3393 O-minimal structures 3404 Local triviality along strata 3415 Mostowski’s Lipschitz stratifications 3426 Consequences of Valette’s triangulation theorem 3447 Appendix. Definition of Lipschitz stratifications 345

References 347

24 Gaffney’s work on equisingularity C. T. C. Wall 350

Introduction 3501 Early results on equisingularity 3512 Notations 3563 Integral closures 3574 Invariants 3625 Criteria for equisingularity 366

xii Contents

6 Recent work 3697 Bibliography of Terry Gaffney’s papers 372

References 374

25 Singularities in algebraic data acquisition Y. Yomdin 378

1 Introduction 3782 Examples of moment inversion 3813 Singularities in reconstruction of linear combinations

of δ-functions 3894 Noisy measurements 393

References 394

Preface

The Workshops on Real and Complex Singularities form a series of biennial

meetings organized by the Singularities group at Instituto de Ciencias Matem-

aticas e de Computacao of Sao Paulo University (ICMC-USP), Brazil. Their

main purpose is to bring together world experts and young researchers in

singularity theory, applications and related fields to report recent achievements

and exchange ideas, addressing trends of research in a stimulating environment.

These meetings started in 1990 following two pioneer symposia on Singular-

ity Theory (in fact the very first talks on Singularities held in Brazil) organized

respectively by G. Loibel and L. Favaro in 1982 and 1988 at ICMC-USP. Since

then, with Maria Aparecida Ruas as a driving force, meetings have taken place

every two years between singularists from around the world who find in Sao

Carlos a centre to interact and develop new ideas.

The meeting held from the 27th of July to the 2nd of August 2008 was

the tenth of these workshops. This was a special occasion, for it was also

dedicated to Maria Aparecida Ruas (Cidinha) and Terence Gaffney on their

60th birthdays.

Cidinha and Terry started their scientific connection in 1976 when she was

a Ph.D. student at Brown University in the U.S.A. At that time Terry held a

position as instructor at that university. Their common interest in singularity

theory brought them together and he became her (very young) thesis supervisor.

Cidinha returned to Brazil in 1980 and joined the Singularities group created

by G. Loibel at the ICMC-USP. Her great capacity and notable enthusiasm

has brought the group to a leading position in the Brazilian mathematical

community.

The mathematical interaction between Cidinha and Terry has had an impor-

tant influence on the development of Singularity Theory at the ICMC-USP.

From the beginning Terry has attended almost all workshops organized by the

xiii

xiv Preface

members of Cidinha’s group, contributing to their research with stimulating

discussions and seminars.

This workshop had a total of about 170 participants from about 15 different

countries. The formal proceedings consisted of 27 plenary talks, 27 ordinary

sessions and 3 poster sessions, with a total of 19 posters. The topics were divided

into six categories: real singularities, classification of singularities, topology

of singularities, global theory of singularities, singularities in geometry, and

dynamical systems.

The Scientific Committee was composed of Lev Birbrair (Universidade

Federal do Ceara, Brazil), Jean-Paul Brasselet (Institut de Mathematiques de

Luminy, France), Goo Ishikawa (Hokkaido University, Japan), Shyuichi Izu-

miya (Hokkaido University, Japan), Steven Kleiman (Massachusetts Institute

of Technology, USA), David Massey (Northeastern University, USA), David

Mond (University of Warwick, UK), Maria del Carmen Romero Fuster (Univer-

sitat de Valencia, Spain), Marcio Gomes Soares (Universidade Federal de Minas

Gerais, Brazil), Marco Antonio Teixeira (Universidade Estadual de Campinas,

Brazil), David Trotman (Universite de Provence, France) and Terry Wall (Uni-

versity of Liverpool, UK).

Thanks are due to many people and institutions crucial in the realization

of the workshop. We start by thanking the Organizing Committee: Roberta

Wik Atique, Abramo Hefez, Isabel Labouriau, Miriam Manoel, Ana Claudia

Nabarro, Regilene Oliveira and Marcelo Jose Saia. We also thank the mem-

bers of the Scientific Committee for their support. The workshop was funded by

FAPESP, CNPq, CAPES, USP and SBM, whose support we gratefully acknowl-

edge. Finally, it is a pleasure to thank the speakers and the other participants

whose presence was the real success of the tenth Workshop.

The editors

Preface xv

Introduction

This book is a selection of papers submitted for the proceedings of the 10th

Workshop on Real and Complex Singularities. They are grouped into three cat-

egories: singularity theory (7 papers), singular varieties (8 papers) and appli-

cations to dynamical systems, generic geometry, singular foliations, etc. (10

papers). Among them, four are survey papers: Local Euler obstruction, old

and new, II, by N. G. Grulha Jr. and J.-P. Brasselet, Global classifications and

graphs, by J. Martınez-Alfaro, C. Mendes de Jesus and M. C. Romero-Fuster,

Pairs of foliations on surfaces, by F. Tari, and Gaffney’s work on equisingular-

ity, by C. T. C. Wall.

We thank the staff members of the London Mathematical Society involved

with the preparation of this book. All papers presented here have been refereed.

1

On a conjecture by A. Durfee

e. artal bartolo, j. carmona ruber

and a. melle-hernandez

Abstract

We show how superisolated surface singularities can be used to

find a counterexample to the following conjecture by A. Durfee [8]:

for a complex polynomial f (x, y, z) in three variables vanishing at

0 with an isolated singularity there, “the local complex algebraic

monodromy is of finite order if and only if a resolution of the

germ ({f = 0}, 0) has no cycles”. A Zariski pair is given whose

corresponding superisolated surface singularities, one has complex

algebraic monodromy of finite order and the other not (answering

a question by J. Stevens).

1. Introduction

In this paper we give an example of a superisolated surface singularity

(V, 0) ⊂ (C3, 0) such that a resolution of the germ (V, 0) has no cycles and the

local complex algebraic monodromy of the germ (V, 0) is not of finite order,

contradicting a conjecture proposed by Durfee [8].

For completeness in the second section we recall well known results about

monodromy of the Milnor fibration, about normal surface singularities and

state the question by Durfee.

In the third section we recall results on superisolated surface singularities

and with them we study in detail the counterexample.

In the last section we show a Zariski pair (C1, C2) of curves of degree

d given by homogeneous polynomials f1(x, y, z) and f2(x, y, z) whose

2000 Mathematics Subject Classification 14B05 (primary), 32S05, 32S10 (secondary).The first author is partially supported through grant MEC (Spain) MTM2007-67908-C02-01. Thelast two authors are partially supported through grant MEC (Spain) MTM2007-67908-C02-02.

1

2 E. Artal Bartolo, J. Carmona Ruber & A. Melle-Hernandez

corresponding superisolated surface singularities (V1, 0) = ({f1(x, y, z) +

ld+1 = 0}, 0) ⊂ (C3, 0) and (V2, 0) = ({f2(x, y, z) + ld+1 = 0}, 0) ⊂ (C3, 0)

(l is a generic hyperplane) satisfy: 1) (V1, 0) has complex algebraic mon-

odromy of finite order and 2) (V2, 0) has complex algebraic monodromy of

infinite order (answering a question proposed to us by J. Stevens).

2. Invariants of singularities

2.1. Monodromy of the Milnor fibration

Let f : (Cn+1, 0) → (C, 0) be an analytic function defining a germ (V, 0) :=

(f −1{0}, 0) ⊂ (Cn+1, 0) of a hypersurface singularity. The Milnor fibration

of the holomorphic function f at 0 is the C∞ locally trivial fibration f | :

Bε(0) ∩ f −1(D∗η) → D∗

η, where Bε(0) is the open ball of radius ε centered

at 0, Dη = {z ∈ C : |z| < η} and D∗η is the open punctured disk (0 < η ≪ ε

and ε small enough). Milnor’s classical result also shows that the topology of

the germ (V, 0) in (Cn+1, 0) is determined by the pair (S2n+1ε , L2n−1

V ), where

S2n+1 = ∂Bε(0) and L2n−1V := S2n+1

ε ∩ V is the link of the singularity.

Any fiber Ff,0 of the Milnor fibration is called the Milnor fiber of f at 0. The

monodromy transformation h : Ff,0 → Ff,0 is the well-defined (up to isotopy)

diffeomorphism of Ff,0 induced by a small loop around 0 ∈ Dη. The complex

algebraic monodromy of f at 0 is the corresponding linear transformation

h∗ : H∗(Ff,0, C) → H∗(Ff,0, C) on the homology groups.

If (V, 0) defines a germ of isolated hypersurface singularity then

Hj (Ff,0, C) = 0 but for j = 0, n. In particular the non-trivial complex alge-

braic monodromy will be denoted by h : Hn(Ff,0, C) → Hn(Ff,0, C) and �V (t)

will denote its characteristic polynomial.

2.2. Monodromy Theorem and its supplements

The following are the main properties of the monodromy operator, see e.g.

[11]:

(a) �V (t) is a product of cyclotomic polynomials.

(b) Let N be the maximal size of the Jordan blocks of h, then N ≤ n + 1.

(c) Let N1 be the maximal size of the Jordan blocks of h for the eigenvalue 1,

then N1 ≤ n.

(d) The monodromy h is called of finite order if there exists N > 0 such that

hN = Id. Le D.T. [12] proved that the monodromy of an irreducible plane

curve singularity is of finite order.

(e) This result was extended by van Doorn and Steenbrink [7] who

showed that if h has a Jordan block of maximal size n + 1 then

On a conjecture by A. Durfee 3

N1 = n, i.e. there exists a Jordan block of h of maximal size n for the

eigenvalue 1.

Milnor proved that the link L2n−1V is (n − 2)-connected. Thus the link is an

integer (resp. rational) homology (2n − 1)-sphere if Hn−1(L2n−1V , Z) = 0 (resp.

Hn−1(L2n−1V , Q) = 0). These can be characterized considering the natural map

h − id : Hn(Ff,0, Z) → Hn(Ff,0, Z) and using Wang‘s exact sequence which

reads as (see e.g. [19, 21]):

0 → Hn

(

L2n−1V , Z

)

→ Hn(Ff,0, Z) → Hn(Ff,0, Z) → Hn−1

(

L2n−1V , Z

)

→ 0.

Thus rank Hn(L2n−1V ) = rank Hn−1(L2n−1

V ) = dim ker(h − id) and:

– L2n−1V is a rational homology (2n − 1)-sphere ⇐⇒ �V (1) �= 0,

– L2n−1V is an integer homology (2n − 1)-sphere ⇐⇒ �V (1) = ±1.

2.3. Normal surface singularities

Let (V, 0) = ({f1 = . . . = fm = 0}, 0) ⊂ (CN , 0) be a normal surface singu-

larity with link LV := V ∩ S2N−1ε , LV is a a connected compact oriented 3-

manifold. Since V ∩ Bε is a cone over the link LV then LV characterizes the

topological type of (V, 0). The link LV is called a rational homology sphere

(QHS) if H1(LV , Q) = 0, and LV is called an integer homology sphere (ZHS)

if H1(LV , Z) = 0. One of the main problems in the study of normal surfaces is

to determine which analytical properties of (V, 0) can be read from the topology

of the singularity, see the very nice survey paper by Nemethi [20].

The resolution graph Ŵ(π ) of a resolution π : V → V allows to relate

analytical and topological properties of V . W. Neumann [22] proved that the

information carried in any resolution graph is the same as the information

carried by the link LV . Let π : V → V be a good resolution of the singular

point 0 ∈ V . Good means that E = π−1{0} is a normal crossing divisor. Let

Ŵ(π ) be the dual graph of the resolution (each vertex decorated with the genus

g(Ei) and the self-intersection E2i of Ei in V ). Mumford proved that the

intersection matrix I = (Ei · Ej ) is negative definite and Grauert proved the

converse, i.e., any such graph comes from the resolution of a normal surface

singularity.

Considering the exact sequence of the pair (V , E) and using I is non-

degenerated then

0 −→ coker I −→ H1(LV , Z) −→ H1(E, Z) −→ 0

and rank H1(E) = rank H1(LV ). In fact LV is a QHS if and only if Ŵ(π ) is a

tree and every Ei is a rational curve. If additionally I has determinant ±1 then

LV is an ZHS.


2.4. Number of cycles in the exceptional set E and

Durfee’s conjecture

In general one gets

rank H1(LV ) = rank H1(Ŵ(π )) + 2∑

i

g(Ei),

where rank H1(Ŵ(π )) is the number of independent cycles of the graph Ŵ(π ).

Let n : E → E be the normalization of E. Durfee showed in [8] that the number

of cycles c(E) in E, i.e. c(E) = rank H1(E) − rank H1(E), does not depend on

the resolution and in fact it is equal to c(E) = rank H1(Ŵ(π )). Therefore, E

contains cycles only when the dual graph of the intersections of the components

contains a cycle. Durfee in [8] proposed the following

Conjecture. For a complex polynomial f (x, y, z) in three variables vanish-

ing at 0 with an isolated singularity there, “the local complex algebraic mon-

odromy h is of finite order if and only if a resolution of the germ ({f = 0}, 0)

has no cycles”.

He showed that the conjecture is true in the following two cases:

(1) if f is weighted homogeneuos (the resolution graph is star-shaped and

therefore its monodromy is finite)

(2) if f = g(x, y) + zn. Using Thom-Sebastiani [27], the monodromy of f is

finite if and only if the monodromy of g is finite. Theorem 3 in [8] proves

that the monodromy of f is of finite order if and only a resolution of f

has no cycles.

2.5. Example (main result)

In this paper we show that the conjecture is not true in general, and for that

we use superisolated surface singularities. Let (V, 0) ⊂ (C3, 0) be the germ of

normal surface singularity defined by f := (xz − y2)3 − ((y − x)x2)2 + z7 =

0. Then the minimal good resolution graph ŴV of (the superisolated singularity)

(V, 0) is

−2Γ

−2V :

−4 −1 −42 −1 −3

−2−2

−2


where every dot denotes a rational non-singular curve with the correspond-

ing self-intersection. Thus the link LV is a rational homology sphere and in

particular this graph is a tree, i.e. it has no cycles. But the complex algebraic

monodromy of f at 0 does not have finite order because there exists a Jordan

block of size 2 × 2 for an eigenvalue �= 1.

3. Superisolated surface singularities

Definition 3.1. A hypersurface surface singularity (V, 0) ⊂ (C3, 0) defined as

the zero locus of f = fd + fd+1 + · · · ∈ C{x, y, z} (where fj is homogeneous

of degree j ) is superisolated, SIS for short, if the singular points of the complex

projective plane curve C := {fd = 0} ⊂ P2 are not situated on the projective

curve {fd+1 = 0}, that is Sing(C) ∩ {fd+1 = 0} = ∅. Note that C must be

reduced.

The SIS were introduced by I. Luengo in [17] to study the µ-constant

stratum. The main idea is that for a SIS the embedded topological type (and the

equisingular type) of (V, 0) does not depend on the choice of fj ’s (for j > d,

as long as fd+1 satisfies the above requirement), e.g. one can take fj = 0 for

any j > d + 1 and fd+1 = ld+1 where l is a linear form not vanishing at the

singular points [18].

3.1. The minimal resolution of a SIS

Let π : V → V be the monoidal transformation centered at the maximal ideal

m ⊂ OV of the local ring of V at 0. Then (V, 0) is a SIS if and only if V is a

non-singular surface. Thus π is the minimal resolution of (V, 0). To construct

the resolution graph Ŵ(π ) consider C = C1 + · · · + Cr the decomposition in

irreducible components of the reduced curve C in P2. Let di (resp. gi) be

the degree (resp. genus) of the curve Ci in P2. Then π−1{0} ∼= C = C1 +

· · · + Cr and the self-intersection of Ci in V is Ci · Ci = −di(d − di + 1), [17,

Lemma 3]. Since the link LV can be identified with the boundary of a regular

neighbourhood of π−1{0} in V then the topology of the tangent cone determines

the topology of the abstract link LV [17].

3.2. The minimal good resolution of a SIS

The minimal good resolution of a SIS (V, 0) is obtained after π by doing the

minimal embedded resolution of each plane curve singularity (C,P ) ⊂ (P2, P ),

P ∈ Sing(C). This means that the support of the minimal good resolution graph


ŴV is the same as the minimal embedded resolution graph ŴC of the projective

plane curve C in P2. The decorations of the minimal good resolution graph ŴV

are as follows:

1) the genus of (the strict transform of) each irreducible component Ci of C

is a birational invariant and then one can compute it as an embedded curve in

P2. All the other curves are non-singular rational curves.

2) Let Cj be an irreducible component of C such that P ∈ Cj and with

multiplicity n ≥ 1 at P . After blowing-up at P , the new self-intersection of the

(strict transform of the) curve Cj in the (strict transform of the) surface V is

C2j − n2. In this way one constructs the minimal good resolution graph Ŵ of

(V, 0).

In particular the theory of hypersurface superisolated surface singularities

“contains” in a canonical way the theory of complex projective plane curves.

Example 3.2. If (V, 0) ⊂ (C3, 0) is a SIS with an irreducible tangent cone

C ⊂ P2 then LV is a rational homology sphere if and only if C is a rational

curve and each of its singularities (C,p) is locally irreducible, i.e a cusp.

Example 3.3. For instance, if f = f6 + z7 is given by the equation f6 =

(xz − y2)3 − ((y − x)x2)2. The plane projective curve C defined by f6 = 0

is irreducible with two singular points: P1 = [0 : 0 : 1] (with a singularity of

local singularity type u3 − v10) and P2 = [1 : 1 : 1] (with a singularity of local

singularity type A2) which are locally irreducible. Let π : X → P2 be the

minimal embedded resolution of C at its singular points P1, P2. Let Ei, i ∈ I,

be the irreducible components of the divisor π−1(f −1{0}).

−2 −2

E1 E2

−4 −1

E3

E6

−1

E9

−3

E7

−2E8E5

E4

−2

−2

The minimal good resolution graph ŴV of the superisolated singularity (V, 0)

is given by

−2 −2V :

−4 −1 −42 −1 −3

−2−2

−2

Γ


3.3. The embedded resolution of a SIS

In [2], the first author has studied, for SIS, the Mixed Hodge Structure of the

cohomology of the Milnor fibre introduced by Steenbrink and Varchenko, [28],

[29]. For that he constructed in an effective way an embedded resolution of a

SIS and described the MHS in geometric terms depending on invariants of the

pair (P2, C).

The first author determined the Jordan form of the complex monodromy on

H2(Ff,0, C) of a SIS. Let �V (t) be the corresponding characteristic polynomial

of the complex monodromy on H2(Ff,0, C). Denote by µ(V, 0) = deg(�V (t))

the Milnor number of (V, 0) ⊂ (C3, 0).

Let �P (t) be the characteristic polynomial (or Alexander polynomial) of

the action of the complex monodromy of the germ (C,P ) on H1(FgP , C),

(where gP is a local equation of C at P and FgP denotes the correspond-

ing Milnor fiber). Let µP be the Milnor number of C at P . Recall that if

nP : CP → (C,P ) is the normalization map then µP = 2δP − (rP − 1), where

δP := dimC nP∗ (OCP )/OC,P and rP is the number of local irreducible compo-

nents of C at P .

Let H be a C-vector space and ϕ : H → H an endomorphism of H . The

i-th Jordan polynomial of ϕ, denoted by �i(t), is the monic polynomial such

that for each ζ ∈ C, the multiplicity of ζ as a root of �i(t) is equal to the

number of Jordan blocks of size i + 1 with eigenvalue equal to ζ .

Let �1 and �2 be the first and the second Jordan polynomials of the

complex monodromy on H2(Ff,0, C) of V and let �P1 be the first Jordan

polynomial of the complex monodromy of the local plane singularity (C,P ).

After the Monodromy Theorem these polynomials joint with �V (t) and �P ,

P ∈ Sing(C), determine the corresponding Jordan form of the complex mon-

odromy. Let us denote the Alexander polynomial of the plane curve C in

P2 by �C(t), it was introduced by A. Libgober [13, 14] and F. Loeser and

Vaquie [16].

Theorem 3.4 [2]. Let (V, 0) be a SIS whose tangent cone C = C1 ∪ . . . ∪ Cr

has r irreducible components and degree d. Then the Jordan form of the complex

monodromy on H2(Ff,0, C) is determined by the following polynomials

(i) The characteristic polynomial �V (t) is equal to

�V (t) =(td − 1)χ (P2\C)

(t − 1)

∏

P∈Sing(C)

�P (td+1).


(ii) The first Jordan polynomial is equal to

�1(t) =1

�C(t)(t − 1)r−1

∏

P∈Sing(C)

�P1 (td+1)�P

(d)(t)

�P1,(d)(t)

3,

where �P(d)(t) := gcd(�P (t), (td − 1)µ

P

) and �P1,(d)(t) := gcd(�P

1 (t),

(td − 1)µP

).

(iii) The second Jordan polynomial is equal to

�2(t) =∏

P∈Sing(C)

�P1,(d)(t).

Corollary 3.5 [2, Corollaire 5.5.4]. The number of Jordan blocks of size 2 for

the eigenvalue 1 of the complex monodromy h is equal to∑

P∈Sing(C)

(rP − 1) − (r − 1). (3.1)

Let Di be the normalization of Di and C the disjoint union of the Di and

n : C → C be the projection map. Thus the first Betti number of C is 2g :=

2∑

i g(Di) and the first Betti number of C is 2g +∑

P∈Sing(C)(rP − 1) − r + 1.

Then∑

P∈Sing(C)(rP − 1) − (r − 1) is exactly the difference between the first

Betti numbers of C and C. In fact this non-negative integer is equal to the first

Betti number of the minimal embedded resolution graph ŴC of the projective

plane curve C in P2, which is nothing but rank H1(ŴV ).

Corollary 3.6. Let (V, 0) be a SIS whose tangent cone C = C1 ∪ . . . ∪ Cr

has r irreducible components. Then the number of independent cycles c(E) =

rank H1(ŴV ) =∑

P∈Sing(C)(rP − 1) − (r − 1).

In particular E has no cycles if and only if∑

P∈Sing(C)(rP − 1) = (r − 1) if

and only if the complex monodromy h has no Jordan blocks of size 2 for the

eigenvalue 1.

Corollary 3.7 [2, Corollaire 4.3.2]. If for every P ∈ Sing(C), the local mon-

odromy of the local plane curve equation gp at P acting on the homology

H1(FgP , C) of the Milnor fibre FgP has no Jordan blocks of maximal size 2 then

the corresponding SIS has no Jordan blocks of size 3.

Corollary 3.8. Let (V, 0) ⊂ (C3, 0) be a SIS with a rational irreducible tan-

gent cone C ⊂ P2 of degree d whose singularities are locally irreducible. Then:

(i) the link LV is a QHS link and E has no cycles,

(ii) the complex monodromy on H2(Ff,0, C) has no Jordan blocks of size 2

for the eigenvalue 1,


(iii) the complex monodromy on H2(Ff,0, C) has no Jordan blocks of size 3.

(iv) The first Jordan polynomial is equal to

�1(t) =1

�C(t)

∏

P∈Sing(C)

gcd(�P (t), (td − 1)µP

).

The proof follows from the previous description and the fact that if every

P ∈ Sing(C) is locally irreducible then by Le D.T. result (see 2.2) the plane

curve singularity has finite order and �P1 (t) = 1.

Corollary 3.9. Let (V, 0) ⊂ (C3, 0) be a SIS whose tangent cone C = C1 ∪

. . . ∪ Cr has r irreducible components. Assume that∑

P∈Sing(C)(rP − 1) =

(r − 1), then:

(i) E has no cycles,

(ii) the complex monodromy on H2(Ff,0, C) has no Jordan blocks of size 2

for the eigenvalue 1,

(iii) the complex monodromy on H2(Ff,0, C) has no Jordan blocks of size 3.

(iv) The first Jordan polynomial is equal to

�1(t) =1

�C(t)(t − 1)r−1

∏

P∈Sing(C)

gcd(�P (t), (td − 1)µP

).

The proof follows from Corollary 3.6 and the part (e) Monodromy

Theorem 2.2.

3.4. The first Jordan polynomial in Example 3.3

As we described above, the plane projective curve C defined by

f6 = (xz − y2)3 − ((y − x)x2)3 = 0 is irreducible, rational and with

two singular points: P1 = [0 : 0 : 1] (with a singularity of local singularity

type u3 − v10) and P2 = [1 : 1 : 1] (with a singularity of local singularity

type A2) which are unibranched. Let π : X → P2 be the minimal embedded

resolution of C at its singular points P1, P2. Let Ei, i ∈ I, be the irreducible

components of the divisor π−1(f −1{0}). For each j ∈ I, we denote by Nj the

multiplicity of Ej in the divisor of the function f ◦ π and we denote by νj − 1

the multiplicity of Ej in the divisor of π∗(ω) where ω is the non-vanishing

holomorpic 2-form dx ∧ dy in C2 = P2 \ L∞. Then the divisor π∗(C) is a

normal crossing divisor. We attach to each exceptional divisor Ei its numerical

data (Ni, νi).


(3,2) (6,3)

E1 E2

(9,4) (30,13)

E3

E6

(6,5)

E9

(2,2)

E7

(3,3)E8E5

E4

(20,9)

(10,5)

Thus �P1 (t) =(t−1)(t30−1)(t3−1)(t10−1)

= φ30φ15φ6 and �P2 (t) =(t−1)(t6−1)(t3−1)(t2−1)

= φ6,

where φk is the k-th cyclotomic polynomial. Thus, by Corollary 3.8, the only

possible eigenvalues of with Jordan blocks of size 2 are the roots of the poly-

nomial �1(t) =φ2

6

�C (t).

The proof of our main result will be finished if we show that the Alexander

polynomial �C(t) = φ6. The Alexander polynomial, in particular of sextics,

has been investigated in detail by Artal [1], Artal and Carmona [3], Degtyarev

[6], Oka [24], Pho [25], Zariski [30] among others. In [23] Corollary 18, I.2, it

is proved that �C(t) = φ6.

Consider a generic line L∞ in P2, in our example the line z = 0 is generic,

and define f (x, y) = f6(x, y, 1). Consider the (global) Milnor fibration given

by the homogeneous polynomial f6 : C3 → C with Milnor fibre F . Randell

[26] proved that �C(t)(t − 1)r−1 is the characteristic polynomial of the alge-

braic monodromy acting on F : T1 : H1(F, C) → H1(F, C).

Lemma 3.10 (Divisibility properties) [13]. The Alexander polinomial

�C(t)(t − 1)r−1 divides∏

P∈Sing(C) �P (t) and the Alexander polynomial at

infinity (td − 1)d−2(t − 1). In particular the roots of the Alexander polynomial

are d-roots of unity.

To compute the Alexander polynomial �C(t) we combined the method

described in [1] with the methods given in [13], [16] and [9].

Consider for k = 1, . . . , d − 1 the ideal sheaf Ik on P2 defined as follows:

– If Q ∈ P2 \ Sing(C) then IkQ = OP2,Q.

– If P ∈ Sing(C) then IkP is the following ideal of OP2,P : if h ∈ OP2,P then

h ∈ IkP if and only if the vanishing order of π∗(h) along each Ei is, at least,

−(νi − 1) + [ kNi

d] (where [.] stands for the integer part of a real number).

For k ≥ 0 the following map

σk : H 0(P2,OP2 (k − 3)) →⊕

P∈Sing(C)

OP2,P /IkP : h �→ (hP + I

kP )P∈Sing(C)

is well defined (up to scalars) and the result of [13] and [16] reinterpreted in

this language as [1] and [9] reads as follows:


Theorem 3.11 (Libgober, Loeser-Vaquie).

�C(t) =

d−1∏

k=1

(�k(t))lk , (3.2)

where �k(t) := (t − exp( 2kπid

)(t − exp(−2kπid

) and lk = dim coker σk

In our case, by the Divisibility properties (Lemma 3.10), �C(t) divides

�P1 (t)�P2 (t) = φ30φ15φ26 . Thus, by Theorem 3.11, we are only interested in

the case k = 1 and 5, �1(t) = �5(t) = φ6 = (t2 − t − 1). In case k = 1, we

have l1 = 0.

In case k = 5, the ideal I5P1

is the following ideal of OP2,P1:

I5P1

= {h ∈ OP2,P1: (π∗h) ≥ E1 + 3E2 + 4E3 + 4E4 + 8E5 + 13E6}

and with the local change of coordinates u = x − y2, w = y, the generators

of the ideal are I5P1

=<uw, u2, w5 > and the dimension of the quotient vector

space OP2,P1/I5

P1is 6. A basis is given by 1, u,w,w2, w3, w4. The ideal

I5P2

= {h ∈ OP2,P2: (π∗h) ≥ 0E7 + 0E8 + E9} = mP2,P2

and the dimension of the quotient vector space OP2,P2/I5

P2is 1. A basis is given

by 1.

If we take as a basis for the space of conics 1, x, y, x2, y2, xy, the map σ5

σ5 : H 0(P2,OP2 (2)) → OP2,P1/I5

P1× OP2,P2

/I5P2

= C6 × C : h �→(

h + I5P1

, h + I5P2

)

is given in such coordinates by (using u = x − y2): σ5(1) =

(1, 0, 0, 0, 0, 0, 1), σ5(x) = (0, 1, 0, 1, 0, 0, 1), σ5(y) = (0, 0, 1, 0, 0, 0, 1),

σ5(x2) = (0, 0, 0, 0, 0, 1, 1), σ5(y2) = (0, 0, 0, 1, 0, 0, 1) and σ5(xy) =

(0, 0, 0, 0, 1, 0, 1).

Therefore σ5 is injective and dim coker σ5 = 7 − 6 + 0 = 1. The key point

is that u /∈ I5P1

.

4. Zariski pairs

Let us consider C ⊂ P2 a reduced projective curve of degree d defined by an

equation fd (x, y, z) = 0. If (V, 0) ⊂ (C3, 0) is a SIS with tangent cone C, then

the link LV of the singularity is completely determined by C. Let us recall,

that LV is a Waldhausen manifold and its plumbing graph is the dual graph

of the good minimal resolution. In order to determine LV we do not need the


embedding of C in P2, but only its embedding in a regular neighborhood. The

needed data can be encoded in a combinatorial way.

Definition 4.1. Let Irr(C) be the set of irreducible components of C. For

P ∈ Sing(C), let B(P ) be the set of local irreducible components of C. The

combinatorial type of C is given by:

– A mapping deg : Irr(C) → Z, given by the degrees of the irreducible com-

ponents of C.

– A mapping top : Sing(C) → Top, where Top is the set of topological types

of singular points. The image of a singular point is its topological type.

– For each P ∈ Sing(C), a mapping βP : T (P ) → Irr(C) such that if γ is a

branch of C at P , then β)(γ ) is the global irreducible component contain-

ing γ .

Remark 1. There is a natural notion of isomorphism of combinatorial types.

It is easily seen that combinatorial type determines and is determined by any

of the following graphs (with vertices decorated with self-intersections):

– The dual graph of the preimage of C by the minimal resolution of Sing′(C).

The set Sing′(C) is obtained from Sing(C) by forgetting ordinary double

points whose branches belong to distinct global irreducible components. We

need to mark in the graph the r vertices corresponding to Irr(C).

– The dual graph of the minimal good resolution of V . Since the minimal

resolution is unique, it is not necessary to mark vertices.

Note also that the combinatorial type determine the characteristic polyno-

mial �V (t) of V (see Theorem 3.4).

Definition 4.2. A Zariski pair is a set of two curves C1, C2 ⊂ P2 with the same

combinatorial type but such that (P2, C1) is not homeomorphic to (P2, C2).

An Alexander-Zariski pair {C1, C2} is a Zariski pair such that the Alexander

polynomials of C1 and C2 do not coincide.

In [2], (see here Theorem 3.4) it is shown that the Jordan form of complex

monodromy of a SIS is determined by the combinatorial type and the Alexander

polynomial of its tangent cone. The first example of Zariski pair was given by

Zariski, [30, 31]; there exist sextic curves with six ordinary cusps. If these cusps

are (resp. not) in a conic then the Alexander polynomial equals t2 − t + 1

(resp. 1). Then, it gives an Alexander-Zariski pair. Many other examples of

Alexander-Zariski pairs have been constructed (Artal [1], Degtyarev [6]).

We state the main result in [2].


Theorem 4.3. Let V1, V2 be two SIS such that their tangent cones form

an Alexander-Zariski pair. Then V1 and V2 have the same abstract topology

and characteristic polynomial of the monodromy but not the same embedded

topology.

Recall that the Jordan form of the monodromy is an invariant of the embed-

ded topology of a SIS (see Theorem 3.4); since it depends on the Alexander

polynomial �C(t) of the tangent cone.

4.1. Zariski pair of reduced sextics with only one singular

point of type A17

Our next Zariski-pair example (C1, C2) can be found in [1, Theoreme 4.4]. The

curves Ci, i = 1, 2 are reduced sextics with only one singular point P of type

A17, locally given by u2 − v18.

(I) the irreducible componentes of C1 are two non-singular cubics. These

cubics meet at only one point P which moreover is an inflection point of each

of the cubics, i.e. the tangent line to the singular point P goes through the

infinitely near points P , P1 and P2 of C1. The equations of C1 are given

for instance by {f1(x, y, z) := (zx2 − y3 − ayz2 − bz3)(zx2 − y3 − ayz2 −

cz3) = 0}, with a, b, c ∈ C generic.

(II) the irreducible componentes of C2 are two non-singular cubics. These

cubics meet at only one point P which is not an inflection point of any of the

cubics, i.e. the tangent line to the singular point P goes through the infinitely

near points P , P1 of C1 but it is not going through P2. The equations of C2

are given for instance by {f2(x, y, z) := (zx2 − y2x − yz2 − a1(z3 − y(xz −

y2)))(zx2 − y2x − yz2 − a2(z3 − y(xz − y2))) = 0} with a1, a2 ∈ C generic.

Consider the superisolated surface singularities (V1, 0) = ({f1(x, y, z) +

l7 = 0}, 0) ⊂ (C3, 0) and (V2, 0) = ({f2(x, y, z) + l7 = 0}, 0) ⊂ (C3, 0) (l is

a generic hyperplane). In both cases the tangent cone has two irreducible

components and it has only one singular point P of local type u2 − v18 and

therefore �P (t) = (t18 − 1)(t − 1)/(t2 − 1) = φ18φ9φ6φ3φ1, where φk is the

k-th cyclotomic polynomial. Thus the number of local branches is 2 and∑

P∈Sing(C)(rP − 1) = (r − 1). By Corollary 3.9, for (Vi, 0), i = 1, 2, the com-

plex monodromy has no Jordan blocks of size 2 for the eigenvalue 1, and it has

no Jordan blocks of size 3. Moreover the first Jordan polynomial is equal to

�1(t) =gcd(�P (t), (t6 − 1)µ

P

)

�Ci(t)(t − 1)

=φ6φ3

�Ci(t)

. (4.1)

To compute �Ci(t) we use the same ideas as in Theorem 3.11.


Lemma 4.4. For the point P at the curve C1 the ideals IkP = OP2,P if k ≤ 3,

I4P =< y3, z > OP2,P and I5

P =< y6, z − y3 − ay4 − by5 > OP2,P .

Lemma 4.5. For the point P at the curve C2 the ideals IkP = OP2,P if k ≤ 3,

I4P =< y3, z − y2 > OP2,P and I5

P =< y6, z − y2 − y5 > OP2,P .

Thus �Ci(t) = φ

dim coker σ5

6 φdim coker σ4

3 .

Therefore the map σ4 is

σ4 : H 0(P2,OP2 (1)) ≃ C3 → OP2,P /I4P ≃ C3,

and if we choose as basis of the first space {1, y, z} and of the second {1, y, y2}

then

(1) by Lemma 4.4, for C1 the dimension dim coker σ4 = dim ker σ4 = 1.


On the other hand for the map σ5

σ5 : H 0(P2,OP2 (2)) ≃ C6 → OP2,P /I5P ≃ C6,

if we choose as basis of the first space {1, y, z, y2, yz, z2} and of the second

{1, y, y2, y3, y4, y5} then we can compute



Therefore, �C1(t) = φ6φ3 and �C2

(t) = 1 and by (4.1) we have proved that

the pair (C1, C2) is a Alexander-Zariski pair.

Example 4.6. Consider the superisolated surface singularities (V1, 0) =

({f1(x, y, z) + l7 = 0}, 0) ⊂ (C3, 0) and (V2, 0) = ({f2(x, y, z) + l7 =

0}, 0) ⊂ (C3, 0) (l is a generic hyperplane). Then the complex algebraic

monodromy of (V1, 0) ⊂ (C3, 0) has finite order and the complex algebraic

monodromy of (V2, 0) ⊂ (C3, 0) has not finite order

This answers a question proposed to us by J. Stevens: find a Zariski pair

C1, C2 such that for the corresponding SIS surface singularities (V1, 0) ⊂

(C3, 0) and (V2, 0) ⊂ (C3, 0) one has a finite order monodromy and the other

it does not.

There are also examples of Zariski pairs which are not Alexander-Zariski

pairs (see [23], [3], [4]). Some of them are distinguished by the so-called

characteristic varieties introduced by Libgober [15]. These are subtori of (C∗)r ,

r := # Irr(C), which measure the excess of Betti numbers of finite Abelian

coverings of the plane ramified on the curve (as Alexander polynomial does it

for cyclic coverings).


Problem 1. How can one translate characteristic varieties of a projective curve

in terms of invariants of the SIS associated to it?

References

1. E. Artal, Sur les couples des Zariski, J. Algebraic Geom., 3 (1994) 223-247.

2. E. Artal, Forme de Jordan de la monodromie des singularites superisolees de sur-

faces, Mem. Amer. Math. Soc. 525 (1994).

3. E. Artal and J. Carmona, Zariski pairs, fundamental groups and Alexander poly-

nomials, J. Math. Soc. Japan 50(3) (1998) 521–543.

4. E. Artal, J. Carmona, J. I. Cogolludo, and H.O. Tokunaga, Sextics with singular

points in special position, J. Knot Theory Ramifications 10 (2001) 547–578.

5. E. Artal, J. I. Cogolludo, and H. O. Tokunaga, A survey on Zariski pairs, Algebraic

geometry in East Asia—Hanoi 2005, 1–100, Adv. Stud. Pure Math. 50 Math. Soc.

Japan, Tokyo, 2008.

6. A. Degtyarev, Alexander polynomial of a curve of degree six, Knot theory and

Ramif. 3(4) (1994) 439–454.

7. M. G M. van Doorn and J. H. M. Steenbrink, A supplement to the monodromy

theorem, Abh. Math. Sem. Univ. Hamburg 59 (1989) 225–233.

8. A. Durfee, The monodromy of a degenerating family of curves, Inv. Math. 28 (1975)

231–241.

9. H. Esnault, Fibre de Milnor d’un cone sur une courbe plane singuliere, Inv. Math.

68 (1982) 477–496.

10. G.-M. Greuel, G. Pfister, and H. Schonemann. Singular 3.0 — A computer algebra

system for polynomial computations. In M. Kerber and M. Kohlhase: Symbolic

computation and automated reasoning, The Calculemus-2000 Symposium (2001),

227–233.

11. A. Landman, On the Picard-Lefschetz transformation for algebraic manifolds

acquiring general singularities, Trans. Amer. Math. Soc. 181 (1973) 89–126.

12. D. T. Le, Sur les noeuds algebriques, Compositio Math. 25 (1972) 281–321.

13. A. Libgober, Alexander polynomial of plane algebraic curves and cyclic multiple

planes, Duke Math. J. 49(4) (1982) 833–851.

14. A. Libgober, Alexander invariants of plane algebraic curves, Singularities, Part 2

(Arcata, Calif., 1981), 135–143, Proc. Sympos. Pure Math. 40 Amer. Math. Soc.,

Providence, RI, 1983.

15. A. Libgober, Characteristic varieties of algebraic curves, Applications of algebraic

geometry to coding theory, physics and computation (Eilat, 2001), Kluwer Acad.

Publ., Dordrecht (2001) 215–254.

16. F. Loeser and M. Vaquie, Le polynome d’Alexander d’une courbe plane projective,

Topology 29(2) (1990) 163–173.

17. I. Luengo, The µ-constant stratum is not smooth, Invent. Math., 90(1) (1987) 139–

152.

18. I. Luengo and A. Melle Hernandez, A formula for the Milnor number C.R. Acad.

Sc. Paris, 321, Serie I. (1995) 1473–1478.

19. A. Nemethi, Some topological invariants of isolated hypersurface singularities, Low

dimensional topology (Eger, 1996/Budapest, 1998), 353–413, Bolyai Soc. Math.

Stud., 8, JUSnos Bolyai Math. Soc., Budapest, 1999.


20. A. Nemethi, Invariants of normal surface singularities, Real and complex singular-

ities, 161–208, Contemp. Math. 354 AMS, Providence, RI, 2004.

21. A. Nemethi and J. H. M. Steenbrink, On the monodromy of curve singularities,

Math. Z. 223(4) (1996) 587–593.

22. W.D. Neumann, A calculus for plumbing applied to the topology of complex

surface singularities and degenerating complex curves, Trans. Amer. Math. Soc.

268(2) (1981) 299–344.

23. M. Oka, A new Alexander-equivalent Zariski pair, Dedicated to the memory of Le

Van Thiem (Hanoi, 1998). Acta Math. Vietnam. 27 (2002) 349–357.

24. M. Oka, Alexander polynomials of sextics, J. Knot Theory Ramifications 12(5)

(2003) 619–636.

25. D.T. Pho, Classification of singularities on torus curves of type (2, 3), Kodai Math.

J., 24 (2001) 259–284.

26. R. Randell, Milnor fibers and Alexander polynomials of plane curves, Singulari-

ties, Part 2 (Arcata, Calif., 1981), 415–419, Proc. Sympos. Pure Math. 40 AMS,

Providence, RI, 1983.

27. M. Sebastiani and R. Thom, Un resultat sur la monodromie, Invent. Math. 13

(1971) 90–96.

28. J. H. M. Steenbrink, Mixed Hodge structure on the vanishing cohomology, Real

and Complex Singularities (Proc. Nordic Summer School, Oslo, 1976) Alphen a/d

Rijn: Sijthoff & Noordhoff (1977) 525–563.

29. A. Varchenko, Asymptotic Hodge structure on vanishing cohomology, Math. USSR

Izvestija 18 (1982) 469–512.

30. O. Zariski, On the problem of existence of algebraic functions of two variables

possessing a given branch curve, Amer. J. Math. 51 (1929) 305–328.

31. O. Zariski, The topological discriminant group of a Riemann surface of genus p,

Amer. J. Math. 59 (1937) 335–358.

E. Artal Bartolo

Departamento de Matematicas, IUMA

Universidad de Zaragoza

Campus Pza. San Francisco s/n

E-50009 Zaragoza SPAIN

[email protected]

A. Melle-Hernandez

Departamento de Algebra,

Universidad Complutense,

Plaza de las Ciencias s/n,

Ciudad Universitaria,

28040 Madrid, SPAIN

[email protected]

J. Carmona Ruber

Departamento de Sistemas

Informaticos y Computacion,

Universidad Complutense,

Plaza de las Ciencias s/n,

Ciudad Universitaria,

28040 Madrid, SPAIN

[email protected]

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